Divide using synthetic division $$ ( 16x^{3}-20x^{2}-4x+15 ) \div ( x+15 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-15&16&-20&-4&15\\& & -240& 3900& \color{black}{-58440} \\ \hline &\color{blue}{16}&\color{blue}{-260}&\color{blue}{3896}&\color{orangered}{-58425} \end{array} $$The solution is:
$$ \frac{ 16x^{3}-20x^{2}-4x+15 }{ x+15 } = \color{blue}{16x^{2}-260x+3896} \color{red}{~-~} \frac{ \color{red}{ 58425 } }{ x+15 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 15 = 0 $ ( $ x = \color{blue}{ -15 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-15}&16&-20&-4&15\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-15&\color{orangered}{ 16 }&-20&-4&15\\& & & & \\ \hline &\color{orangered}{16}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -15 } \cdot \color{blue}{ 16 } = \color{blue}{ -240 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-15}&16&-20&-4&15\\& & \color{blue}{-240} & & \\ \hline &\color{blue}{16}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ \left( -240 \right) } = \color{orangered}{ -260 } $
$$ \begin{array}{c|rrrr}-15&16&\color{orangered}{ -20 }&-4&15\\& & \color{orangered}{-240} & & \\ \hline &16&\color{orangered}{-260}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -15 } \cdot \color{blue}{ \left( -260 \right) } = \color{blue}{ 3900 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-15}&16&-20&-4&15\\& & -240& \color{blue}{3900} & \\ \hline &16&\color{blue}{-260}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 3900 } = \color{orangered}{ 3896 } $
$$ \begin{array}{c|rrrr}-15&16&-20&\color{orangered}{ -4 }&15\\& & -240& \color{orangered}{3900} & \\ \hline &16&-260&\color{orangered}{3896}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -15 } \cdot \color{blue}{ 3896 } = \color{blue}{ -58440 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-15}&16&-20&-4&15\\& & -240& 3900& \color{blue}{-58440} \\ \hline &16&-260&\color{blue}{3896}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ \left( -58440 \right) } = \color{orangered}{ -58425 } $
$$ \begin{array}{c|rrrr}-15&16&-20&-4&\color{orangered}{ 15 }\\& & -240& 3900& \color{orangered}{-58440} \\ \hline &\color{blue}{16}&\color{blue}{-260}&\color{blue}{3896}&\color{orangered}{-58425} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 16x^{2}-260x+3896 } $ with a remainder of $ \color{red}{ -58425 } $.